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  2. Real number - Wikipedia

    en.wikipedia.org/wiki/Real_number

    There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. [19]

  3. Subset - Wikipedia

    en.wikipedia.org/wiki/Subset

    These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...

  4. Least-upper-bound property - Wikipedia

    en.wikipedia.org/wiki/Least-upper-bound_property

    Every non-empty subset of the real numbers which is bounded from above has a least upper bound. In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers.

  5. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...

  6. Infimum and supremum - Wikipedia

    en.wikipedia.org/wiki/Infimum_and_supremum

    The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset of the real numbers has an infimum and a supremum. If S {\displaystyle S} is not bounded below, one often formally writes inf S = − ∞ . {\displaystyle \inf _{}S=-\infty .}

  7. Dense set - Wikipedia

    en.wikipedia.org/wiki/Dense_set

    In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine ...

  8. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    The real number system is the unique complete ordered field, ... For subsets of the real numbers, there are several equivalent definitions of compactness.

  9. Neighbourhood (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Neighbourhood_(mathematics)

    The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M. Given the set of real numbers with the usual Euclidean metric and a subset defined as := (; /), then is a neighbourhood for the set of natural numbers, but is not a uniform neighbourhood of this set.